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Colouring Paths in Directed Symmetric Trees with Applications to WDM Routing
, 1997
"... . Let T be a symmetric directed tree, i.e., an undirected tree with each edge viewed as two opposite arcs. We prove that the minimum number of colours needed to colour the set of all directed paths in T , so that no two paths of the same colour use the same arc of T , is equal to the maximum number ..."
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Cited by 16 (1 self)
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. Let T be a symmetric directed tree, i.e., an undirected tree with each edge viewed as two opposite arcs. We prove that the minimum number of colours needed to colour the set of all directed paths in T , so that no two paths of the same colour use the same arc of T , is equal to the maximum number of paths passing through an arc of T . This result is applied to solve the alltoall communication problem in wavelength divisionmultiplexing (WDM) routing in alloptical networks, that is, we give an efficient algorithm to optimally assign wavelengths to the all the paths of a tree network. It is known that the problem of colouring a general subset of all possible paths in a symmetric directed tree is an NPhard problem. We study conditions for a given set S of paths be coloured efficiently with the minimum possible number of colours/wavelengths. 1 Introduction Let T be a tree and x; y two vertices of T . The dipath P (x; y) in T is the undirected path joining x to y, in which each ed...
Routing In AllOptical Networks: Algorithmic And GraphTheoretic Problems
 Information and Complexity
, 2000
"... : This paper surveys theoretical results for wavelengthrouting in all optical networks and presents several open problems. We focus our attention on graphtheoretical problems and proof techniques. 1 INTRODUCTION Optical networks are emerging as key technology in communication networks and are ..."
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Cited by 12 (2 self)
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: This paper surveys theoretical results for wavelengthrouting in all optical networks and presents several open problems. We focus our attention on graphtheoretical problems and proof techniques. 1 INTRODUCTION Optical networks are emerging as key technology in communication networks and are expected to dominate many applications, such as video conferencing, scientific visualisation, realtime medical imaging, highspeed supercomputing and distributed computing [18, 19, 33, 36]. The books of Green [18] and McAulay [30] offer a comprehensive overview of the physical theory and applications of this emerging technology. In WDM (Wavelength Division Multiplexing) optical networks, the bandwidth available in optical fiber is utilised by partitioning it into several channels, each at a different wavelength. Each wavelength can carry a separate stream of data. In general, a WDM network consists of routing nodes interconnected by pointtopoint unidirectional optic fiber links. Each...
On the Physical and Logical Topology Design of LargeScale Optical Networks
 JOURNAL OF LIGHTWAVE TECHNOLOGY
, 2003
"... We consider the problem of designing a network of optical crossconnects (OXCs) to provide endtoend lightpath services to large numbers of label switched routers (LSRs). We present a set of heuristic algorithms to address the combined problem of physical topology design (i.e., determine the number ..."
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We consider the problem of designing a network of optical crossconnects (OXCs) to provide endtoend lightpath services to large numbers of label switched routers (LSRs). We present a set of heuristic algorithms to address the combined problem of physical topology design (i.e., determine the number of OXCs required and the fiber links among them) and logical topology design (i.e., determine the routing and wavelength assignment for the lightpaths among the LSRs). Unlike previous studies which were limited to small topologies with a handful of nodes and a few tens of lightpaths, we have applied our algorithms to networks with hundreds or thousands of LSRs and with a number of lightpaths that is an order of magnitude larger than the number of LSRs. In order to characterize the performance of our algorithms, we have developed lower bounds which can be computed efficiently. We present numerical results for up to 1000 LSRs and for a wide range of system parameters such as the number of wavelengths per fiber, the number of transceivers per LSR, and the number of ports per OXC. The results indicate that it is possible to build largescale optical networks with rich connectivity in a costeffective manner, using relatively few but properly dimensioned OXCs.
AlltoAll Routing and Coloring in Weighted Trees of Rings
 IN PROCEEDINGS OF ELEVEN ANNUAL ACM SYMPOSIUM ON PARALLEL ALGORITHMS AND ARCHITECTURES
, 1999
"... A tree of rings is an undirected graph obtained from the union of rings, which intersect two by two in at most one node, such that any two nodes are connected by exactly two edgedisjoint paths. In this paper, we consider symmetric directed trees of rings with weighted nodes. A routing for a weighte ..."
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Cited by 7 (0 self)
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A tree of rings is an undirected graph obtained from the union of rings, which intersect two by two in at most one node, such that any two nodes are connected by exactly two edgedisjoint paths. In this paper, we consider symmetric directed trees of rings with weighted nodes. A routing for a weighted digraph is a collection of directed paths (dipaths), such that for each ordered pair of nodes (x 1 ; x 2 ) with respective weights w 1 and w 2 , there are w 1 w 2 dipaths (possibly not distinct) from x 1 to x 2 . Motivated by the Wavelength Division Multiplexing (WDM) technology in alloptical networks, we study the problem of nding a routing which can be colored by the fewest number of colors so that dipaths of the same color are arcdisjoint. We prove that this minimum number of colors (wavelengths) is equal to the maximum number of dipaths that share one arc (load), minimized over all routings. The problem can be efficiently solved (dipaths found and colored) using cut properties.
Optical AlltoAll Communication for Some Product Graphs (Extended Abstract)
 PROC. OF THE 24TH SEMINAR ON CURRENT TRENDS IN THEORY AND PRACTICE OF INFORMATICS, SOFSEM'97, LECTURE NOTES IN COMPUTER SCIENCE 1338
, 1997
"... The problem of alltoall communication in a network consists of designing directed paths between any ordered pair of vertices in a symmetric directed graph and assigning them minimum number of colours such that every two dipaths sharing an edge have distinct colour. We prove several exact results o ..."
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Cited by 7 (3 self)
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The problem of alltoall communication in a network consists of designing directed paths between any ordered pair of vertices in a symmetric directed graph and assigning them minimum number of colours such that every two dipaths sharing an edge have distinct colour. We prove several exact results on the number of colours for some Cartesian product graphs, including 2dimensional (toroidal) square meshes of odd side, which completes previous results for even sided square meshes.
Vertex Disjoint Routings of Cycles over Tori
, 2007
"... We study the problem of designing a survivable WDM network based on covering the communication requests with subnetworks that are protected independently from each other. We consider here the case when the physical network is T (n), a torus of size n by n, the subnetworks are cycles and the communic ..."
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Cited by 6 (0 self)
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We study the problem of designing a survivable WDM network based on covering the communication requests with subnetworks that are protected independently from each other. We consider here the case when the physical network is T (n), a torus of size n by n, the subnetworks are cycles and the communication scheme is alltoall or total exchange (where all pairs of vertices communicate). We will represent the communication requests by a logical graph: a complete graph for the scheme of alltoall. This problem can be modeled as follows: find a cycle partition or covering of the request edges of Kn2, such that for each cycle in the partition, its request edges can be routed in the physical network T (n) by a set of vertex disjoint paths (equivalently, the routings with the request cycle form an elementary cycle in T (n)). Let the load of an edge of the WDM network be the number of paths associated with the requests using the edge. The cost of the network depends on the total load (the cost of transmission) and the maximum load (the cost of equipment). To minimize these costs, we will search for an optimal (or quasi optimal) routing satisfying the following two conditions: (a) each request edge is routed by a shortest path over T (n), and (b) the load of each physical edge resulting from the routing of all cycles of S is uniform or quasi uniform. In this paper, we find a covering or partition of the request edges of K n 2 into cycles with an associated optimal or quasi optimal routing such that either (1) the number of cycles of the covering is minimum, or (2) the cycles have size 3 or 4.
On the design of multifiber WDM networks
 IN PROC. ALGOTEL'02
, 2001
"... In this paper, we address multifiber optical networks with Wavelength Division Multiplexing (wdm). Assuming that the lightpaths use the same wavelength from source to destination, we extend the definition of the wellknown Wavelength Assignment Problem (wap), to the case where there are k fibers p ..."
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Cited by 6 (1 self)
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In this paper, we address multifiber optical networks with Wavelength Division Multiplexing (wdm). Assuming that the lightpaths use the same wavelength from source to destination, we extend the definition of the wellknown Wavelength Assignment Problem (wap), to the case where there are k fibers per link, and w wavelengths per fiber are available. We then develop a new model for the (k; w)wap, based on conflict hypergraphs : Conflict hypergraphs more accurately capture the lightpath interdependencies, generalizing the conflict graphs used for singlefiber networks. By relating the (k; w)wap with the hypergraph coloring problem, we prove that the former is NPcomplete, and present further results with respect to the complexity of that problem. Finally, we analyze the practical performances of two methodologies based on hypergraph coloring, on existing backbone networks in Europe and in the USA. The first relies on an integer programming formulation and the second consists of a heuristic based on a randomized algorithm. We consider the two natural optimization problems that arise from the (k; w)wap: the problem of minimizing k given w, and that of minimizing w given k.
AlltoAll Optical Routing in Chordal Rings of Degree Four
, 1999
"... We consider the problem of routing in networks employing alloptical routing technology. In such networks, information between nodes of the network is transmitted as light on fiberoptic lines without being converted to electronic form in between. We consider switched optical networks that use th ..."
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Cited by 6 (3 self)
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We consider the problem of routing in networks employing alloptical routing technology. In such networks, information between nodes of the network is transmitted as light on fiberoptic lines without being converted to electronic form in between. We consider switched optical networks that use the wavelengthdivision multiplexing (or WDM) approach. A WDM network consists of nodes connected by pointtopoint fiberoptic links, each of which can support a fixed number of wavelengths. The switches are capable of redirecting incoming streams based on wavelengths, without changing the wavelengths. Different messages may use the same link concurrently if they are assigned distinct wavelengths. However, messages assigned the same wavelength must be assigned edgedisjoint paths. Given a communication instance in a network, the optical routing problem is the assignment of routes to communication requests of the instance, as well as wavelengths to routes so that the number of wavelen...
ON fWISE ARC FORWARDING INDEX AND WAVELENGTH ALLOCATIONS IN FAULTY Alloptical Hypercubes
, 2003
"... ..."
Permutation Communication in AllOptical Rings
, 1998
"... We study the wavelength problem and arc (edge) congestion problem for communicating permutation instances on a ring. We prove the best possible upper bounds on the number of wavelengths and arc (edge) congestion in both directed and undirected cases. ..."
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Cited by 5 (1 self)
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We study the wavelength problem and arc (edge) congestion problem for communicating permutation instances on a ring. We prove the best possible upper bounds on the number of wavelengths and arc (edge) congestion in both directed and undirected cases.